Domain and range of Integrals of functions $x^2/(2x)$ and $x/2$?

Question

I have two functions one $f(x)= \frac{x^2}{2x}$ and $g(x)=x/2$. Are their integrals F(x) and G(x) 100% same or is there a difference? Something that I don't understand is this: the two functions domains are different when $x=0$? What about their derivatives and integrals? Please explain it briefly.

Answer 1

Two functions are identical if they have the same values for every input and they have the same inputs--the same domain.

Your two functions are almost identical; you can turn one easily into the other, but there is a technicality. You aren't allowed to divide by $0$. So one of the functions fails to exist at $0$ and the other exists there and has a value. They are almost but not quite the same function.

However, in order for their areas to be different, you would need a finite area discrepancy and a single point does not prove that. Functions that are equal "almost everywhere" have the same integrals. So these two functions have exactly the same definite integrals. Their indefinite integrals are different at a single point, because those are functions, and also any two antiderivatives (indefinite integrals) can be off by a constant value.

As for derivatives, remember that you can't take the derivative of a function if it is not continuous there, and a function can't be continuous if it doesn't exist. So there is a difference between their derivatives, but only at a single point.